left exact functor and exact sequence

1. Proposition

Let 20231215-covariant_left_exact_functor_and_exact_sequence_8bca0a22355f86c50b3e315a050fae8717462859.svg be abelian categories and be an Ab-enriched-functor (see: ) TFAE:

is a left exact functor
for an exact sequence , the induced sequence

20231215-covariant_left_exact_functor_and_exact_sequence_b1c1bf38077295535c0d1efe7e8f914024fcb8f6.svg

is also exact

2. Proof

kernel and exactness

2.1. 1) 2)

Suppose

20231215-covariant_left_exact_functor_and_exact_sequence_dd2df6f06603bd398bce100ecccc6e52c98c9d49.svg

is an exact sequence.

Then as an additive functor preserves the zero object, we get the sequence

where by assumption 20231215-covariant_left_exact_functor_and_exact_sequence_f8b6cbb83c12b0e9e1222af231db2f8212dbcbe9.svg is a kernel of . therefore the sequence is exact.

left exact functor preserves zero morphisms

2.2. 2) 1)

Consider as exact sequence

20231215-covariant_left_exact_functor_and_exact_sequence_746948bedaa371e18eec693ad32d899aee08adcb.svg

Then

20231215-covariant_left_exact_functor_and_exact_sequence_65273b44f90365685173a43c96e196b3bb36a4f2.svg

is also exact

after applying 20231215-covariant_left_exact_functor_and_exact_sequence_f1aeb1ac71cce52f443892bfa958e0fd4a349772.svg we get

20231215-covariant_left_exact_functor_and_exact_sequence_42a097c4dd9887276f64e795cd412f0b185af207.svg

as exact sequence and hence by kernel and exactness it follows that

20231215-covariant_left_exact_functor_and_exact_sequence_e347cf0e4697c76c985e35e098b273bd0d9b6667.svg

is a kernel of 20231215-covariant_left_exact_functor_and_exact_sequence_b055af7d9a64c5ff8704462b5dc2f57813ec72bd.svg .

Here 20231215-covariant_left_exact_functor_and_exact_sequence_f1aeb1ac71cce52f443892bfa958e0fd4a349772.svg preserves monomorphism